Re: Open problems

[Michael Barr <barr@math.mcgill.ca>]

I just wanted to add to what Vaughan said about the provenance of Chu 
spaces.  When I started working on duality in 1975 at the ETH, one 
prominent example was topologicalvector spaces.  So I looked at several 
books on the subject, one by Grothendieck.  They all mentioned a 
construction of pairs (V,V') of spaces equipped with a bilinear map V x V' 
--> K (the ground field).  This simple example, with its obvious duality 
led pretty directly to the Chu construction.  It was amazing that it 
turned to not only give and obvious duality but also a natural 
*-autonomous structure (that Peter Chu studied).

At some point I decided to track down the source.  It certainly had an air 
of Grothendieck about it, but no, it turned out to go back to George 
Mackey's thesis, published in TAMS around 1944, I think.  He looked at 
pairs (V,V') as above and assumed they were extensional but not 
necessarily separated.  He was aware of the duality in the separated 
extensional case, but didn't pursue it in any detail.

So what was the motivation.  I can guess that he was thinking that V', as 
a space of functionals on V, coud replace the topology.  Just as in a 
locally compact abelian group, you could recover the topology if you knew 
which characters were "admissible".  I once wrote him to ask if that was 
his original motivation, but he died within the ensuing year, not having 
answered me.

But truly, it goes back to Mackey.

Michael

----- Original Message -----
From: "Vaughan Pratt" <pratt@cs.stanford.edu>
To: "Categories mailing list" <categories@mta.ca>
Sent: Sunday, 14 December, 2014 1:12:58 PM
Subject: categories: Re: Open problems

On 12/12/2014 1:56 PM, Harley Eades III wrote:
> Dee Roytenberg?s email pushed me to write an email I have been wanting to write
> for a bit.
>
> One thing I have been trying to do recently is figure out what the major (and minor)
> open problems are with respect to applications of category theory to CS.   I am very
> new to this area, and getting an idea of what folks are working on, and what problems
> people feel are important will help young researchers learns where to concentrate
> their efforts.

At http://thue.stanford.edu/puzzle.html can be seen a problem about Chu
spaces that's been open for close to two decades, and that's been
translated during the past three years into Belorussian, Ukrainian,
Polish, Slovenian, Czech, French, and Bulgarian by volunteers who
apparently found the problem sufficiently interesting to be worth the
effort, perhaps motivated by its connection to crossword puzzles.

The problem is whether every T1 comonoid in Chu(Set,2) is discrete.

This is so for two special cases.

1. Countable comonoids, taking cardinality of a Chu space to be that of
its points.

2. Representable comonoids (of any cardinality), in the sense of
representation by directed CPOs as per the hierarchy of full embeddings
on page 6 of the paper "Comonoids in chu: a large cartesian closed
sibling of topological spaces" at
http://boole.stanford.edu/pub/comonoids.pdf .  T1 DCPOs are of course
discrete simply because T1 posets are discrete and the structure of any
DCPO is determined by its order via the Scott topology (as distinct from
the Alexandroff topology, though they coincide in the T1 case).

The problem remains open for uncountable comonoids not representable as
DCPOs.

...


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